"This city is my home."
Translation:Этот город — мой дом.
это means 'that/this is'.
You need to use the masculine demonstrative pronoun этот.
1) "Города" = "cities" in the nominative and the accusative cases. 2) "This is my home." = "Это мой дом.", but when "this" is before a noun then its translation depends on a gender of the translated noun. "Город" is masculine, therefore "Этот город - мой дом."
Example: "This ring is my precious". The "ring" in Russian has a neutral gender, the "precious" as a noun in Russian has a feminine gender, therefore "Это кольцо - моя прелесть.".*
*This translation of "my precious" ("моя прелесть") is traditional in Russia for films "The Lord of the Rings" and memes.
You will want to read 'A Guide to Using ЭТО', by olimo, found here: >> https://www.duolingo.com/comment/11536858
As far as my incomplete understanding of the subject goes, it comes down to "это <noun>" meaning "the <noun>" and "этот <noun>" meaning "This particular variant of <noun>," where этот can be replaced by the particular version required for amount(singular/plural) and gender(male/female/neutral).
Do read the guide, it is far more helpful than any single comment I can give you.